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For the given matrix \( B = \begin{bmatrix} 4 & 1 \\ 2 & 3 \end{bmatrix} \):

1. The eigenvalues of matrix \( B \) are:
   - \( \lambda_1 = 5 \)
   - \( \lambda_2 = 2 \)

2. The corresponding eigenvectors for each eigenvalue are:
   - For \( \lambda_1 = 5 \), the eigenvector is \( \begin{bmatrix} 1 \\ 1 \end{bmatrix} \)
   - For \( \lambda_2 = 2 \), the eigenvector is \( \begin{bmatrix} -\frac{1}{2} \\ 1 \end{bmatrix} \)

Would you like more details or have any questions?

Here are 5 related questions you might explore:
1. How do you verify the correctness of eigenvalues and eigenvectors?
2. What is the significance of eigenvalues in practical applications?
3. Can you calculate the determinant of matrix \( B \) using its eigenvalues?
4. What does it mean if a matrix has repeated eigenvalues?
5. How do eigenvectors relate to diagonalization of a matrix?

**Tip:** Eigenvectors corresponding to distinct eigenvalues are always linearly independent.

Given the following matrix B = [[4, 1], [2, 3]], find the eigenvalues of matrix B and the corresponding eigenvectors for each eigenvalue.

Learn how to find the eigenvalues and corresponding eigenvectors for matrix B = [[4, 1], [2, 3]]. This problem covers key concepts in linear algebra such as solving for eigenvalues using the determinant formula and finding eigenvectors. Suitable for undergraduate students in linear algebra.

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